Changes

===Commentary Note 11.18===

===Commentary Note 11.18===

There is a comment that I ought to make on the concept of a ''structure preserving map'', including as a special case the idea of an ''order-preserving map''.  It seems to be a peculiarity of mathematical usage in general — at least, I don't think it's just me — that "preserving structure" always means "preserving ''some'', not of necessity ''all'' of the structure in question".  People sometimes express this by speaking of ''structure preservation in measure'', the implication being that any property that is amenable to being qualified in manner is potentially amenable to being quantified in degree, perhaps in such a way as to answer questions like "How structure-preserving is it?".

An ''order-preserving map'' is a special case of a ''structure preserving map'', and the idea of ''preserving structure'', as used in mathematics, always means preserving ''some'' but not necessarily ''all'' the structure of the source domain in question.  People sometimes express this by speaking of ''structure preservation in measure'', the implication being that any property that is amenable to being qualified in manner is potentially amenable to being quantified in degree, perhaps in such a way as to answer questions like "How structure-preserving is it?".

Let's see how this remark applies to the order-preserving property of the "number of" mapping ''v''&nbsp;:&nbsp;''S''&nbsp;&rarr;&nbsp;'''R'''.  For any pair of absolute terms ''x'' and ''y'' in the syntactic domain ''S'', we have the following implications, where "–<" denotes the logical subsumption relation on terms and "=<" is the "less than or equal to" relation on the real number domain R.

Let's see how this remark applies to the order-preserving property of the "number of" mapping <math>v : S \to \mathbb{R}.</math> For any pair of absolute terms <math>x\!</math> and <math>y\!</math> in the syntactic domain <math>S,\!</math> we have the following implications, where <math>^{\backprime\backprime}-\!\!\!<\!^{\prime\prime}</math> denotes the logical subsumption relation on terms and <math>^{\backprime\backprime}\!\!\le\!^{\prime\prime}</math> denotes the ''less than or equal to'' relation on the real number domain <math>\mathbb{R}.</math>

: ''x'' –< ''y'' &rArr; ''vx'' =< ''vy''

: ''x'' –< ''y'' &rArr; ''vx'' =< ''vy''